The Stacks project

7.49 Sheafification in a topology

In this section we explain the analogue of the sheafification construction in a topology.

Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of sets. For every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define

\[ L\mathcal{F}(U) = \mathop{\mathrm{colim}}\nolimits _{S \in J(U)^{opp}} \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) \]

as a colimit. Here we think of $J(U)$ as a partially ordered set, ordered by inclusion, see Lemma 7.47.2. The transition maps in the system are defined as follows. If $S \subset S'$ are in $J(U)$, then $S \to S'$ is a morphism of presheaves. Hence there is a natural restriction mapping

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S', \mathcal{F}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}). \]

Thus we see that $S \mapsto \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})$ is a directed system as in Categories, Definition 4.21.2 provided we reverse the ordering on $J(U)$ (which is what the superscript ${}^{opp}$ is supposed to indicate). In particular, since $h_ U \in J(U)$ there is a canonical map

\[ \ell : \mathcal{F}(U) \longrightarrow L\mathcal{F}(U) \]

coming from the identification $\mathcal{F}(U) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F})$. In addition, the colimit defining $L\mathcal{F}(U)$ is directed since for any pair of covering sieves $S, S'$ on $U$ the sieve $S \cap S'$ is a covering sieve too, see Lemma 7.47.2.

Let $f : V \to U$ be a morphism in $\mathcal{C}$. Let $S \in J(U)$. There is a commutative diagram

\[ \xymatrix{ S \times _ U V \ar[r] \ar[d] & h_ V \ar[d] \\ S \ar[r] & h_ U } \]

We can use the left vertical map to get canonical restriction maps

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}) \to \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U V, \mathcal{F}). \]

Base change $S \mapsto S \times _ U V$ induces an order preserving map $J(U) \to J(V)$. And the restriction maps define a transformation of functors as in Categories, Lemma categories-lemma-functorial-colimit. Hence we get a natural restriction map

\[ L\mathcal{F}(U) \longrightarrow L\mathcal{F}(V). \]

Lemma 7.49.1. In the situation above.

  1. The assignment $U \mapsto L\mathcal{F}(U)$ combined with the restriction mappings defined above is a presheaf.

  2. The maps $\ell $ glue to give a morphism of presheaves $\ell : \mathcal{F} \to L\mathcal{F}$.

  3. The rule $\mathcal{F} \mapsto (\mathcal{F} \xrightarrow {\ell } L\mathcal{F})$ is a functor.

  4. If $\mathcal{F}$ is a subpresheaf of $\mathcal{G}$, then $L\mathcal{F}$ is a subpresheaf of $L\mathcal{G}$.

  5. The map $\ell : \mathcal{F} \to L\mathcal{F}$ has the following property: For every section $s \in L\mathcal{F}(U)$ there exists a covering sieve $S$ on $U$ and an element $\varphi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})$ such that $\ell (\varphi )$ equals the restriction of $s$ to $S$.

Proof. Omitted. $\square$

Definition 7.49.2. Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. We say that a presheaf of sets $\mathcal{F}$ is separated if for every object $U$ and every covering sieve $S$ on $U$ the canonical map $\mathcal{F}(U) \to \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})$ is injective.

Theorem 7.49.3. Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. Let $\mathcal{F}$ be a presheaf of sets.

  1. The presheaf $L\mathcal{F}$ is separated.

  2. If $\mathcal{F}$ is separated, then $L\mathcal{F}$ is a sheaf and the map of presheaves $\mathcal{F} \to L\mathcal{F}$ is injective.

  3. If $\mathcal{F}$ is a sheaf, then $\mathcal{F} \to L\mathcal{F}$ is an isomorphism.

  4. The presheaf $LL\mathcal{F}$ is always a sheaf.

Proof. Part (3) is trivial from the definition of $L$ and the definition of a sheaf (Definition 7.47.10). Part (4) follows formally from the others.

We sketch the proof of (1). Suppose $S$ is a covering sieve of the object $U$. Suppose that $\varphi _ i \in L\mathcal{F}(U)$, $i = 1, 2$ map to the same element in $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, L\mathcal{F})$. We may find a single covering sieve $S'$ on $U$ such that both $\varphi _ i$ are represented by elements $\varphi _ i \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S', \mathcal{F})$. We may assume that $S' = S$ by replacing both $S$ and $S'$ by $S' \cap S$ which is also a covering sieve, see Lemma 7.47.2. Suppose $V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\alpha : V \to U$ in $S(V)$. Then we have $S \times _ U V = h_ V$, see Lemma 7.47.5. Thus the restrictions of $\varphi _ i$ via $V \to U$ correspond to sections $s_{i, V, \alpha }$ of $\mathcal{F}$ over $V$. The assumption is that there exist a covering sieve $S_{V, \alpha }$ of $V$ such that $s_{i, V, \alpha }$ restrict to the same element of $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S_{V, \alpha }, \mathcal{F})$. Consider the sieve $S''$ on $U$ defined by the rule

7.49.3.1
\begin{eqnarray} \label{sites-equation-S-prime-prime} (f : T \to U) \in S''(T) & \Leftrightarrow & \exists \ V , \ \alpha : V \to U, \ \alpha \in S(V), \nonumber \\ & & \exists \ g : T \to V, \ g \in S_{V, \alpha }(T), \\ & & f = \alpha \circ g \nonumber \end{eqnarray}

By axiom (2) of a topology we see that $S''$ is a covering sieve on $U$. By construction we see that $\varphi _1$ and $\varphi _2$ restrict to the same element of $\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S'', L\mathcal{F})$ as desired.

We sketch the proof of (2). Assume that $\mathcal{F}$ is a separated presheaf of sets on $\mathcal{C}$ with respect to the topology $J$. Let $S$ be a covering sieve of the object $U$ of $\mathcal{C}$. Suppose that $\varphi \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(S, L\mathcal{F})$. We have to find an element $s \in L\mathcal{F}(U)$ restricting to $\varphi $. Suppose $V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $\alpha : V \to U$ in $S(V)$. The value $\varphi (\alpha ) \in L\mathcal{F}(V)$ is given by a covering sieve $S_{V, \alpha }$ of $V$ and a morphism of presheaves $\varphi _{V, \alpha } : S_{V, \alpha } \to \mathcal{F}$. As in the proof above, define a covering sieve $S''$ on $U$ by Equation (7.49.3.1). We define

\[ \varphi '' : S'' \longrightarrow \mathcal{F} \]

by the following simple rule: For every $f : T \to U$, $f \in S''(T)$ choose $V, \alpha , g$ as in Equation (7.49.3.1). Then set

\[ \varphi ''(f) = \varphi _{V, \alpha }(g). \]

We claim this is independent of the choice of $V, \alpha , g$. Consider a second such choice$ V', \alpha ', g'$. The restrictions of $\varphi _{V, \alpha }$ and $\varphi _{V', \alpha '}$ to the intersection of the following covering sieves on $T$

\[ (S_{V, \alpha } \times _{V, g} T) \cap (S_{V', \alpha '} \times _{V', g'} T) \]

agree. Namely, these restrictions both correspond to the restriction of $\varphi $ to $T$ (via $f$) and the desired equality follows because $\mathcal{F}$ is separated. Denote the common restriction $\psi $. The independence of choice follows because $\varphi _{V, \alpha }(g) = \psi (\text{id}_ T) = \varphi _{V', \alpha '}(g')$. OK, so now $\varphi ''$ gives an element $s \in L\mathcal{F}(U)$. We leave it to the reader to check that $s$ restricts to $\varphi $. $\square$

Definition 7.49.4. Let $\mathcal{C}$ be a category endowed with a topology $J$. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. The sheaf $\mathcal{F}^\# := LL\mathcal{F}$ together with the canonical map $\mathcal{F} \to \mathcal{F}^\# $ is called the sheaf associated to $\mathcal{F}$.

Proposition 7.49.5. Let $\mathcal{C}$ be a category endowed with a topology. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. The canonical map $\mathcal{F} \to \mathcal{F}^\# $ has the following universal property: For any map $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf of sets, there is a unique map $\mathcal{F}^\# \to \mathcal{G}$ such that $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ equals the given map.

Proof. Same as the proof of Proposition 7.10.12. $\square$

Comments (2)

Comment #5861 by Tathagata Basak on

There seems to be a typo in the second displayed equation. Given covering sieves , the natural restriction mapping should go the other way.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00ZG. Beware of the difference between the letter 'O' and the digit '0'.



View Section 7.49 as pdf